Local Blow-up Research Articles

On the local existence and blow-up solutions to a quasi-linear bi-hyperbolic equation with dynamic boundary conditions

This note aims to study the existence of the local solutions and derive a blow-up result for a quasi-linear bi-hyperbolic equation with dynamic boundary conditions. We use the maximal monotone operator theory to demonstrate the solution’s local well-posedness, and a concavity method to establish the blow-up result.

Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

<p>In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a $ L^p-L^q $ estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.</p>

Local well-posedness and blow-up criterion to a nonlinear shallow water wave equation

<abstract><p>The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained.</p></abstract>

On a logarithmic wave equation with nonlinear dynamical boundary conditions: local existence and blow-up

This paper deals with a hyperbolic-type equation with a logarithmic source term and dynamic boundary condition. Given convenient initial data, we obtained the local existence of a weak solution. Local existence results of solutions are obtained using the Faedo-Galerkin method and the Schauder fixed-point theorem. Additionally, under suitable assumptions on initial data, the lower bound time of the blow-up result is investigated.

Uniqueness, multiplicity and nondegeneracy of positive solutions to the Lane-Emden problem

In this paper, we study the nearly critical Lane-Emden equations(⁎){−Δu=up−εinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN with N≥3, p=N+2N−2 and ε>0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions.In general, the solutions of (⁎) may blow-up at multiple points a1,⋯,ak of Ω as ε→0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k=1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎).

On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise

The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving gradient operator. Then, we obtain results on local solution, blow-up criterion, and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data is also studied for the multiplicative noise case.

Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity

In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are Lp(Lq)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.

Local Existence and Blow-Up of Solutions for Wave Equation Involving the Fractional Laplacian with Nonlinear Source Term

The aim of this paper is to investigate the local weak existence and vacuum isolating of solutions, asymptotic behavior, and blow-up of the solutions for a wave equation involving the fractional Laplacian with nonlinear source. By means of the Galerkin approximations, we prove the local weak existence and finite time blow-up of the solutions and we give the upper and lower bounds for blow-up time.

Sub-diffusion equations with Mittag-Leffler nonlinearity

<p style='text-indent:20px;'>The paper is devoted to the study of subdiffusion equations with Mittag-Leffler nonlinearity. The comparison principle is proved in a bounded domain. The results on the local existence, global existence, and blow-up of solutions to the initial-boundary value problem are obtained. In addition, the results of local existence and blow-up of solutions to the initial problem on the whole Euclidean space are proven.</p>

Einstein vacuum equations with 𝕌(1) symmetry in an elliptic gauge: Local well-posedness and blow-up criterium

In this paper, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying [Formula: see text] symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the [Formula: see text] level. In addition, we prove a blow-up criterium at the [Formula: see text] level. By doing so, we improve a result of Huneau and Luk in [Einstein equations under polarized [Formula: see text] symmetry in an elliptic gauge, Commun. Math. Phys. 361(3) (2018) 873–949] on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms.

Global well-posedness, regularity and blow-up for the β-CCF model

The paper is concerned with the β-CCF model which is a 1D nonlocal nonlinear transport model. The velocity is coupled via a composition of the Hilbert transform and Riesz potentials, giving rise to three basic cases that reflect the balance between the nonlinearity and dissipation, namely the subcritical, critical and supercritical ranges. In our analysis we consider those three cases and obtain a set of results containing local well-posedness, global smoothness, eventual regularity and blow-up of solutions.

Wave breaking phenomenon for the generalized FORQ equation

In this paper, we consider the generalized Fokas–Olver–Rosenau–Qiao equation (also called the ab-family of equation). The local well-posedness and blow-up scenario are investigated. Sufficient conditions on the initial data to guarantee wave breaking are established for b = 2. For b = 2 and 1/6 < a ≤ 1/3, we also show that if the initial momentum density is positive in a compact support and the length of the support is small enough, then wave breaking occurs.

LOCAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO FUJITA-TYPE EQUATIONS INVOLVING GENERAL ABSORPTION TERM ON FINITE GRAPHS

This paper is devoted to the study of the behaviors of the solution to Fujita-type equations on finite graphs. Under certain conditions given by absorption term of the equations, we prove respectively local existence and blow-up results of solutions to Fujita-type equations on finite graphs. Our results contain some previous results as special cases. Finally, we provide some numerical experiments to illustrate the applicability of the obtained results.

Local well-posedness and finite time blow-up of solutions to an attraction–repulsion chemotaxis system in higher dimensions

We consider the Cauchy problem for an attraction–repulsion chemotaxis system in Rn with the chemotactic coefficients of the attractant β1 and the repellent β2. In particular, these coefficients are important role in the global existence and blow up of the solutions. In this paper, we show the local well-posedness of solutions in the critical spaces Ln/2(Rn) and the finite time blow-up of the solution under the condition β1>β2 in higher dimensional spaces.

Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient

<p style='text-indent:20px;'>We consider the fourth-order Schrödinger equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \alpha>0, \mu = \pm1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M2">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lambda\in\mathbb{C} $\end{document}</tex-math></inline-formula>. Firstly, we prove local well-posedness in <inline-formula><tex-math id="M4">\begin{document}$ H^4\left( {\mathbb R}^N\right) $\end{document}</tex-math></inline-formula> in both <inline-formula><tex-math id="M5">\begin{document}$ H^4 $\end{document}</tex-math></inline-formula> subcritical and critical case: <inline-formula><tex-math id="M6">\begin{document}$ \alpha>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ (N-8)\alpha\leq8 $\end{document}</tex-math></inline-formula>. Then, for any given compact set <inline-formula><tex-math id="M8">\begin{document}$ K\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula>, we construct <inline-formula><tex-math id="M9">\begin{document}$ H^4( {\mathbb R}^N) $\end{document}</tex-math></inline-formula> solutions that are defined on <inline-formula><tex-math id="M10">\begin{document}$ (-T, 0) $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M11">\begin{document}$ T>0 $\end{document}</tex-math></inline-formula>, and blow up exactly on <inline-formula><tex-math id="M12">\begin{document}$ K $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M13">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>.</p>

A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise

We establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in Hilbert spaces. The key requirement relies on an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. By establishing a cancellation estimate for certain differential operators of order one with suitable coefficients, we give the detailed constructions of such regular approximations for certain examples. In particular, we show novel local-in-time results for the stochastic two-component Camassa–Holm system and for the stochastic Córdoba–Córdoba–Fontelos model.

Non-degeneracy of single-peak solutions to a Kirchhoff equation

ABSTRACT In this paper, we are concerned with the following singular perturbation Kirchhoff equation where constants and 1<p<5. By using local Pohozaev identity and blow-up analysis, we derive the non-degeneracy of the single-peak solutions to the above problem under some suitable condition on for small ε.

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

Uniqueness of single-peaked solutions to singularity perturbed semilinear Dirichlet problems

We are concerned with the uniqueness of single-peaked solutions to the following problem −ε2△u+u=Q(x)up−1,inΩ,u>0,inΩ,u=0,on∂Ω,where ε>0 is a small parameter, and N≥3, 2<p<2N∕(N−2). By local Pohozaev identity and blow-up analysis, we show the uniqueness of single-peaked solutions under certain assumptions on asymptotic behavior of Q(x) and its first derivatives near the critical point. Here the degeneracy of the critical point is also allowed, which gives a partial answer to the question proposed in Cao et al. (1998).

On the local existence and blow-up for generalized SQG patches

We study patch solutions of a family of transport equations given by a parameter \(\alpha \), \(0< \alpha <2\), with the cases \(\alpha =0\) and \(\alpha =1\) corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for \(H^{2}\) patches in the half-space setting for \(0<\alpha < 1/3\), allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of \(\alpha \) for which finite time singularities have been shown in Kiselev et al. (Commun Pure Appl Math 70(7):1253–1315, 2017) and Kiselev et al. (Ann Math 3:909–948, 2016). Second, we establish that patches remain regular for \(0<\alpha <2\) as long as the arc-chord condition and the regularity of order \(C^{1+\delta }\) for \(\delta >\alpha /2\) are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in Córdoba et al. (Proc Natl Acad Sci USA 102:5949–5952, 2005) and Scott and Dritschel (Phys Rev Lett 112:144505, 2014) for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in Gancedo (Adv Math 217(6):2569–2598, 2008) and in Chae et al. (Commun Pure Appl Math 65(8):1037–1066, 2012), giving local existence for patches in \(H^{2}\) for \(0<\alpha < 1\) and in \(H^3\) for \(1<\alpha <2\).